发表机构
Anhui Science and Technology University（安徽科技学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了一个Apéry型序列的两步超同余猜想，通过超几何转换、差分算子谱分析及有限域恒等式，给出了更强的赋值结果，并利用有限计算处理例外素数。
AI 中文摘要
设$G_n=\sum_{k=0}^n4^k\binom{2n-2k}{n-k}^{2}\binom{2k}{k}$。孙智宏猜想：对于素数$p\equiv3\pmod4$、正奇数$m$以及$r\ge2$，两步同余$G_{(mp^r-1)/2}\equiv p^2G_{(mp^{r-2}-1)/2}\pmod {p^{2r-1}}$成立。我们证明了更强的赋值陈述：对每个正奇数$M$，有$G_{(p^2M-1)/2}-p^2G_{(M-1)/2}\in p^{2v_p(M)+3}\mathbb Z_p$。证明将该和式转化为一个终止的${}_3F_2$超几何级数，构造了一个消去数字转移算子，并识别出其三次差分算子的二维解析商。该商算子的特征多项式为$X^2-p^2$；其第二迹通过Gross--Koblitz公式和Greene的有限域Dixon恒等式求值。一个对数损失的Green逆将该谱恒等式转化为实际的积分解析原函数。例外素数$3$需要有限精确的PARI/GP证书，而无限尾部则以符号方式界定。因此，该计算是有限的、可复现的，并且与证明的一致部分相分离。
英文摘要
Let $G_n=\sum_{k=0}^n4^k\binom{2n-2k}{n-k}^{2}\binom{2k}{k}$. Zhi-Hong Sun conjectured that, for primes $p\equiv3\pmod4$, positive odd integers $m$, and $r\ge2$, the two-step congruence $G_{(mp^r-1)/2}\equiv p^2G_{(mp^{r-2}-1)/2}\pmod {p^{2r-1}}$ holds. We prove the stronger valuation statement $G_{(p^2M-1)/2}-p^2G_{(M-1)/2}\in p^{2v_p(M)+3}\mathbb Z_p$ for every positive odd $M$. The proof converts the sum to a terminating ${}_3F_2$, constructs a cancelled digit-transfer operator, and identifies a two-dimensional analytic quotient of its cubic difference operator. The quotient operator has characteristic polynomial $X^2-p^2$; its second trace is evaluated through the Gross--Koblitz formula and Greene's finite-field Dixon identity. A logarithmic-loss Green inverse converts this spectral identity into an actual integral analytic primitive. The exceptional prime $3$ requires a finite exact PARI/GP certificate, while the infinite tail is bounded symbolically. Thus the computation is finite, reproducible, and separated from the uniform part of the proof.
Comments23 pages. Ancillary files contain the exact PARI/GP certificate for the exceptional prime 3 and a fail-closed runner
